# Optimum Sum Codes, that Effectively Detect the Errors of Low Multiplicities

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Optimum Sum Codes, that Effectively Detect the Errors of Low Multiplicities
Valery Sapozhnikov, Vladimir Sapozhnikov, Dmitry Efanov, Vyacheslav Dmitriev, Maria Cherepanova
Automation and Remote Control on Railways Department, Petersburg State Transport University, Russian Federation
Abstract — The article provides the method of formation of the sum code with minimum total number of undetectable errors. The code, suggested by authors, has the same number of check bits, as a classic Berger code, but also has a better detection ability, particularly within the area of low multiplicity errors in data vectors. New sum code allows to develop concurrent error detection (CED) systems of logic units of automation and computer devices with low equipment redundancy and a high percentage of error detection in controlled blocks.
I. Introduction
Sum codes are often used for data transmission and processing systems, as well as for design of reliable systems of automation and computer devices [1−9]. One of the examples of application of sum codes is a concurrent error detection (CED) system of arithmetical-logical units, that make a part of any modern systems of automation control [10−14].
The CED system structure is presented in Fig. 1. There, the initial arithmetical-logical unit F (x), that realizes the system of operational Boolean functions f (x), f2(x), …, fm (x), is equipped with the special control equipment. The control equipment includes the reference logic block G (x), calculating the values of test functions gi (x), g2(x), …, gk (x), and the self-checking checker, that registers the conformance of the values of operational and test functions at any given time. This conformance is established at the CED system design stage and usually determined according to the rules of formation of preselected sum code. So the block F (x) outputs are matched with the data vector of length m, and the block G (x) outputs — with the check vector of length k. Those significant parameters of the CED system as detection ability and equipment redundancy are substantially depend on the rules of code formation, and the latter, in its turn, effects the power consumption, processing speed, testability and other characteristics of built discrete system [15,16].
The structure, shown in Fig. 1, in practice is built under condition of 100% detection of single faults [1,17]. By the
Valery Sapozhnikov is with the Petersburg State Transport University, Automation and Remote Control on Railways Department, Russian Federation.
Vladimir Sapozhnikov is with the Petersburg State Transport University, Automation and Remote Control on Railways Department, RF.
Dmitry Efanov is with the Petersburg State Transport University, Automation and Remote Control on Railways Department, RF (corresponding author to provide e-mail: mitriche@yandex. ru).
Vyacheslav Dmitriev is with the Petersburg State Transport University, Automation and Remote Control on Railways Department, RF.
Maria Cherepanova is with the Petersburg State Transport University, Automation and Remote Control on Railways Department, RF.
reason of separate implementation of the blocks F (x), G (x) and checker at any given time the fault can occur only in one element of CED system. The reference logic block faults distort the values of test functions, that is registered by the checker. The checker, as the watchdog in CED system, is built as a self-checking device and thus detects its own faults at leas in one input set [1]. The CED system task is to provide the detection of single faults in block F (x), when the internal configuration of links between logic elements within it can result in occurrence of distortions of different multiplicities in the values of output vector. Therefore, it is possible to consider the characteristics of sum code based on error detection in data vectors, studying by this the features of the CED system itself.
Different sum codes have different characteristics of error detection in data vectors and allow to build systems with various technical specifications. This paper is dedicated to the study of sum codes, that have minimum total number of undetectable errors in data vectors, as well as decreased number of double undetectable errors in comparison with classic sum codes.
II. Analysis of Berger codes and sum codes with
WEIGHTED TRANSITIONS PROPERTIES FOR ERROR DETECTION IN DATA VECTORS
Classic sum code, or Berger code [18], is built based on the following principle: data vector weight r (the sum of one data bits), and then the obtained value is presented in binary form and recorded into the bits of the check vector. The number of check bits in Berger code is calculated using the formula: k = [log2(m + 1)] (notation is an integer, upward to the calculating value). Hereafter referred the Berger code to as S (m, k)-code.
In S (m, k)-code the same check vector corresponds to all data vectors with the same weight. For this value r is Cm of
data vectors. With increasing of r the number of data vectors, corresponding to the same check vector, increases with
its maximum at m for the even m and at m -1 for the odd
2 2
m, and then grows away. Such distribution of data vectors between the check vectors results in the fact, that S (m, k)-code does not detect significantly large amount of errors in data vectors. For example, S (5,3)-code does not detect 220 errors in data vectors, that forms 22. 18% of all possible errors in data vectors (i.e. this code does not detect slightly less than one fourth of errors in data vectors). With low value of detection ability all S (m, k)-codes also have low efficiency of low multiplicities error detection — these codes do not detect 50% of double errors, 37. 5% of quadruple errors, 31. 25% six-fold errors, etc. [19].
Data bits
Inputs
Operational
outputs
Check
signal
Fig. 1. CED system structure
So the problem of increasing the detection abiliity arises, which includes the detection ability within the area of low multiplicity errors, on condition of keeping the number of check bits in the code. Practically speaking, for example, in CED systems the solution of this problem will allow to detect the bigger number of single faults of controlled logic unit without increasing the equipment redundancy.
In [20] the term of an optimum separable code by the criterion of minimum number of undetectable errors for the specified values of m and k was introduced. The optimum code has a uniform distribution of all 2m data vectors among 2k check vectors and does not detect
Nmy = 2m (2m-k — 1) (1)
errors in data vectors. For instance, the optimum code, where m=5 and k=3, has N™n = 25 (25−3 -1)= 32 • 3 = 96
undetectable errors, i.e. 2. 29 times less than S (5,3)-code. The same paper suggests the method for improving the S (m, k)-codes characteristics for error detection in data vectors without sacrificing the number of check bits. This method is based on transformation of each Berger code word in its check part using the specific algorithm. This algorithm implicates the determination of the vector weight to the modulus M = 2log2 (m+1)^-1 and using the correction factor, equals to modulo two sum of values of some data vector bits. Modified Berger codes, at an average, detect twice as many errors in data vectors as classic S (m, k)-codes, however, they are not the optimum codes. In [21−23] there is a suggestion of the line of modified Berger codes, that are obtained by establishing the modulus for data vector weight determination, selected from the set M є {2−4-… -2^log2(m+1)^-2}. These codes have decreased number of check bits in comparison with Berger codes. Modu-lar-modified Berger codes with its m and k values are also not an optimum (excluding the code, where M=2, that has 2 check bits).
Optimum sum code with the same number of check bits as Berger code, can be built by alteration of the algorithm of formation of the sum code with weighted transitions [24−26]:
Algorithm 1. Check vectors obtaining for sum code with weighted transitions:
1. Data vector bits are considered, that have the adjacent positions.
2. Each transition from the bit to the bit it is assigned the weight ratio from the positive integers (1, 2, …, m-1), starting from the lower order bit.
3. The sum of so called active transitions (transition between the bit is called active, if the modulo two sum of the values of adjacent bits equals 1) is calculated.
4. Obtained value is presented in binary form and recorded in check vector.
Let us set the sum code with weighted transitions as WT (m, k)-code.
WT (m, k)-code detects bigger number of errors in data vectors, than S (m, k)-code, and however it has the increased number of check bits. This number is equals
k =
log2
m (m -1) 2
The increased number of check bits
in WT (m, k)-code in comparison with S (m, k)-code results in increase of complexity of control equipment in CED system (see Fig. 1). Besides, the check bits of WT (m, k)-code are used ineffectively — WT (m, k)-codes are not optimum codes [27]. Optimum code are obtained by using the following algorithm of formation.
Algorithm 2. Check vectors obtaining for modified sum code with weighted transitions:
1. The steps 1 — 3 of Algorithm 1 are carried out.
2. The modulus M = 2log2 (m+1)^ value is established (this is the modulus of Berger code).
3. The sum V of so called active transitions is calculated.
4. The least non-negative residue of V value for established modulus is calculated: W = (V) modM.
5. The number W is presented in binary form and recorded in check vector.
To demonstrate the operation of algorithm of formation of modified sum code with weighted transitions or WTM (m, k)-code, let us use the example of WT8(5,3)-code formation. Table I presents the weight ratios of each transition between the bits of data vector, and Table 2 presents all
code words of WT8(5,3)-code. For instance, in & lt-101>- data vector transitions are considered active between the bits x1 and x2, x2 and x3, x3 and x4. To obtain V value it is necessary to sum up the weight ratios of these transitions: V=w1,2+w2,3+w3,4=1+2+3=6. The least non-negative residue of V equals to W=(6)mod8=6, that in binary form is presented as & lt-110>-.
TABLE I
Weight ratios of transitions in data vector
W4.5 W3,4 W2,3 W1,2
4 3 2 1
By analyzing Table II, it is not too difficult to notice that V values for data vectors, equidistant form the middle of the table (these vectors have opposite values of similar bits), are equal. At that the number of repeating V values is unequal (Table III). This results in the irregularity in the distribution of data vectors of WT (m, k)-code among the check vectors (within the test groups). Implementing the modification of WT (m, k)-code under Algorithm 2, this disadvantage is eliminated, because the calculation is made up to the value M-1=7: (8)mod8=0, (9)mod8=l и (10)mod8=2.
TABLE II
CODE WORDS OF WT8(5,3)-CODE
№ Data vector bits V W=(V)mod8 Check vector bits
X5 X4 X3 X2 X] У3 У2 У1
0 0 0 0 0 0 0 0 0 0 0
1 0 0 0 0 1 1 1 0 0 1
2 0 0 0 1 0 3 3 0 1 1
3 0 0 0 1 1 2 2 0 1 0
4 0 0 1 0 0 5 5 1 0 1
5 0 0 1 0 1 6 6 1 1 0
6 0 0 1 1 0 4 4 1 0 0
7 0 0 1 1 1 3 3 0 1 1
8 0 1 0 0 0 7 7 1 1 1
9 0 1 0 0 1 8 0 0 0 0
10 0 1 0 1 0 10 2 0 1 0
11 0 1 0 1 1 9 1 0 0 1
12 0 1 1 0 0 6 6 1 1 0
13 0 1 1 0 1 7 7 1 1 1
14 0 1 1 1 0 5 5 1 0 1
15 0 1 1 1 1 4 4 1 0 0
16 1 0 0 0 0 4 4 1 0 0
17 1 0 0 0 1 5 5 1 0 1
18 1 0 0 1 0 7 7 1 1 1
19 1 0 0 1 1 6 6 1 1 0
20 1 0 1 0 0 9 1 0 0 1
21 1 0 1 0 1 10 2 0 1 0
22 1 0 1 1 0 8 0 0 0 0
23 1 0 1 1 1 7 7 1 1 1
24 1 1 0 0 0 3 3 0 1 1
25 1 1 0 0 1 4 4 1 0 0
26 1 1 0 1 0 6 6 1 1 0
27 1 1 0 1 1 5 5 1 0 1
28 1 1 1 0 0 2 2 0 1 0
29 1 1 1 0 1 3 3 0 1 1
30 1 1 1 1 0 1 1 0 0 1
31 1 1 1 1 1 0 0 0 0 0
TABLE III
DISTRIBUTION OF V VALUES FOR WT8(5,3)-CODE FORMATION
V
0 1 2 3 4 5 6 7 8 9 10
0 1 2 3 4 5 6 7 8 9 10
3 4 5 6 7
3 4 5 6 7
Table Iv provides the distribution of data vectors within the test groups for the WT8(5,3)-code of interest. Within each test group there are 4 data vectors. What is more, in virtue of the equality of V values for equidistant from the middle of Table II data vectors within each test group there are always two data vectors minimum, that differ from each other in all bits. This induces the undetectable errors of maximum multiplicity d=m in WT8(5,3)-code. Undetectable errors of lower multiplicities appear in WT8(5,3)-codes in case of distortions, that transfer data vectors of one test group into data vectors, not-equidistant from the middle of Table II. This type of distribution of data vectors among the test groups remains unchanged and for all WTM (m, k)-codes. The only exclusion is WTM (m, k)-codes with the number of data bits m = 2' (t=1, 2, …). For these codes there is no filling the last test group, because the number of data bits is that the value W = 2k -1 does not form. Actually, where m=4, for example, maximum data vector weight is W=w 1,2+W2,3+W3,4= 1+2+3=6.
Apropos WTM (m, k)-codes with data vectors lengths m Ф 2'- (t=1, 2, .), it should be noticed that within each test group there are transitions for them similar by multiplicity. These transitions determine the number of distortions with different multiplicity within each test group. For example, Fig. 2 shows the graph of all possible transitions within & lt-111>- test group of WT8(5,3)-code.
… d=5 ,…
•: 1 000 *->: 10 111: ¦
d=2
d=2
¦: 1 101 *->: 10 010: ¦
¦…¦ d=5 ¦… '-
Fig. 2. Transitions within & lt-111>- test group of WT8(5,3)-code
The total number of undetectable errors in WTM (m, k)-codes is obtained by multiplying the number of errors within one test group by M.
Table V5 provides the distribution of undetectable errors in data vectors of WTM (m, k)-codes for several m values. Analysis of such table allowed to establish the features of a new class of sum codes based on error detection in data vectors:
— any WTM (m, k)-code detects 100% of single distortions (is a fault-tolerant code) —
— WTM (m, k)-codes with data vector lengths m Ф 2'- ('-=1, 2, .) are optimum-
— WTM (m, k)-codes with data vector lengths m = 2'- ('-=1, 2, .) are close to optimum, but not such codes-
— WTM (m, k)-codes with even values of m detect 100% of errors of odd multiplicities in data vectors (any Berger codes have this feature) —
— WTM (m, k)-codes detect 100% of errors of multiplicity d=m-1 in data vectors-
— WTM (m, k)-codes do not detect 100% of errors of maximum multiplicity d=m in data vectors-
— the number of undetectable errors of each multiplicity (as well as the total number of undetectable errors) is multiple of modulus M.
The properties of WTM (m, k)-codes are explained by the rules of its formation. For example, fault-tolerance is explained by the fact, that for weight ratios the sequence of positive integers from 1 to m-1 is used, and modulus for residue determination is always bigger than the maximum transition weight in data vector: wm-1m=m-1,
M = 2lo%1 (m+1)^ and M& gt-m-1. If modulus M will be equal to
the weight of transition between the highest bits wmm-1, so the residue of this value would be 0: (wm-1m)modM=(M)modM=0. And this would mean that even with activation of transition the weight values in total sum W would always be 0, and the bit would be uncontrolled.
TABLE IV
Distribution of data vectors of WT8(5,3)-code within test groups
W=(V)mod8
0 1 2 3 4 5 6 7
Check vector
000 001 010 011 100 101 110 111
0 1 11 10 110 100 101 1 000
1 001 1 011 1 010 111 1 111 1 110 1 100 1 101
10 110 10 100 10 101 11 000 10 000 10 001 10 011 10 010
11 111 11 110 11 000 11 101 11 001 11 011 11 010 10 111
Table V
Distribution of undetectable errors in data vectors of some WTM (m, k)-codes
m к M Distribution of undetectable errors by multiplicities, Nm, d Nm
1 2 3 4 5 6 7 8 9 10 11 12
2 2 4 0 4 4
3 2 4 0 0 8 8
4 3 8 0 8 0 16 24
5 3 8 0 32 32 0 32 96
6 3 8 0 192 0 192 0 64 448
7 3 8 0 448 448 448 448 0 128 1920
8 4 16 0 832 0 1936 0 832 0 256 3856
9 4 16 0 2304 1280 4096 4096 1280 2304 0 512 15 872
10 4 16 0 7680 0 24 064 0 24 064 0 7680 0 1024 64 512
11 4 16 0 17 408 7424 58 496 45 696 45 696 58 496 7424 17 408 0 2048 260 096
12 4 16 0 44 032 0 242 688 0 466 944 0 242 688 0 44 032 0 4096 1 044 480
Fig. 3. Part of double undetectable errors in total number of double errors in S (m, k) and WTM (m, k)-codes
Fig. 3 gives the comparison of WTM (m, k)-codes and S (m, k)-codes by its ability of detection of double errors in data vectors — there is a function of the number of double undetectable errors in data vectors to the total number of double errors in data vectors (pd) by the data vector length. Based on Fig. 3 it follows that WTM (m, k)-codes more than twice as effective in double errors detection, as S (m, k)-codes. Such improvement is true and for errors of any even multiplicities (Table VI). Diagrams of Fig. 4 show the function of pm, which equals to the number of undetectable errors of given multiplicity in Berger codes
and in weight-based codes of interest, to the data vector length m. The value nm demonstrates what fold decrease the number of undetectable errors of given multiplicity in WTM (m, k)-codes in comparison with S (m, k)-codes with the specified m value.
For odd m values the efficiency of detection of errors of even multiplicities in WTM (m, k)-codes, in comparison with m±1, increases, that happens due to the presence of errors of odd multiplicities within the class of undetectable errors in codes with odd m values.
Fig. 4. Improving the characteristic of double errors detection of WTM (m, k)-codes in comparison with S (m, k)-codes
TABLE VI
Value bd for each multiplicity for classic and weight-based sum codes
m Part of undetectable errors of multiplicity d in total number of errors of given multiplicity, %
1 | 2 I 3 | 4 | 5 | 6 I 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20
WTM (m, k)-code
2 0 100
3 0 0 100
4 0 8. 33 0 100
5 0 10 10 0 100
6 0 20 0 20 0 100
7 0 16. 67 10 10 16. 67 0 100
8 0 11. 61 0 10.8 0 11. 61 0 100
9 0 12.5 2. 98 6. 35 6. 35 2. 98 12.5 0 100
10 0 16. 67 0 11. 19 0 11. 19 0 16. 67 0 100
11 0 15. 45 2.2 8. 66 4. 83 4. 83 8. 66 2.2 15. 45 0 100
12 0 16. 29 0 11. 97 0 12. 34 0 11. 97 0 16. 29 0 100
13 0 15. 54 2. 05 8. 89 4. 75 6. 54 6. 54 4. 75 8. 89 2. 05 15. 54 0 100
14 0 16. 35 0 12. 76 0 12. 27 0 12. 27 0 12. 76 0 16. 35 0 100
15 0 15.6 3.1 9. 23 4. 26 7. 32 5. 79 5. 79 7. 32 4. 26 9. 23 3.1 15.6 0 100
16 0 11. 88 0 6. 61 0 6. 15 0 6. 15 0 6. 15 0 6. 61 0 11. 88 0 100
17 0 12. 87 0. 81 5. 65 1.7 4. 09 2. 49 3. 27 3. 27 2. 49 4. 09 1.7 5. 65 0. 81 12. 87 0 100
18 0 14. 71 0 7. 25 0 6. 26 0 6. 15 0 6. 15 0 6. 26 0 7. 25 0 14. 71 0 100
19 0 14. 33 0. 68 6. 47 1. 46 4.5 2.2 3. 61 2.9 2.9 3. 61 2.2 4.5 1. 46 6. 47 0. 68 14. 33 0 100
20 0 14. 61 0 7. 47 0 6. 43 0 6.2 0 6. 16 0 6.2 0 6. 43 0 7. 47 0 14. 61 0 100
S (m, k)-code
2+20 | 0 | 50 | 0 | 37.5 I 0 I 31. 25 1 0 I 27. 34 I 0 I 24. 61 I 0 I 22. 56 I 0 I 20. 95 I 0 I 19. 64 I 0 I 18. 55 I 0 I 17. 62
III. Conclusion
This paper suggests the method for formation of a sum code with minimum total number of undetectable errors in data vectors. Moreover, this new sum code more effectively detects the errors of low multiplicities in data vectors, than the classic Berger code. These advantages of WTM (m, k)-code can be considered while CED systems design with 100% detection of single faults in controlled logic units.
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